Measures of Central Tendency Test

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Measures of Central Tendency Test - 50

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Weekly Quiz Competition

Question 1
1 / -0

The exam scores of all 500 students were recorded and it was determined that these scores were normally distributed. If Jane's score is 0.8 standard deviation above the mean, then how many, to the nearest unit, students scored above Jane?

A
$$109$$

B
$$106$$

C
$$150$$

D
$$160$$

Solution

Let m be the mean and s be the standard deviation and find the z score.
$$z = (x - m) /s = (0.8 s + m - m) / s = 0.8$$
The percentage of student who scored above Jane is (from table of normal distribution).
1 - 0.7881 = 0.2119 = 21.19%
The number of student who scored above Jane is (from table of normal distribution).
21.19% 0f 500 = 106
Question 2
1 / -0

The AM of first $$5$$ whole numbers is:

A
$$1$$

B
$$2$$

C
$$3$$

D
$$5$$

Solution

The first $$5$$ whole numbers are $$0, 1, 2, 3, 4$$.

We know that,
$$Arithmetic\ mean= \dfrac{Sum\ of\ given\ numbers}{Total\ numbers}$$

Apply the above formula, we get

$$Arithmetic\ mean= \dfrac{0+1+2+3+4}{5}$$
$$= \dfrac{10}{5}$$
$$= 2$$

Hence, option $$(b)$$ is correct.
Question 3
1 / -0

The ages of 40 girls in a class are given below:
Age (in years) 14 15 16 17 18
Number of girls 5 8 15 10 2
Find the mean age.

A
13.9

B
14.9

C
15.9

D
16.9

Solution
Question 4
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Which measures of central tendency get affected if the extreme observations on both the ends of a data arranged in descending order are removed ?

A
Mean and mode

B
Mean and Median

C
Mode and Median

D
Mean,Median and Mode

Solution

Mean and Mode are affected if the observation of the ends are removed.
Median Is not affected as the observation a the middle remains at the middle.

Hence, option (A) is the correct answer.
Question 5
1 / -0

In the formula $$x=a+h(f_i u_i/ f_i)$$, for finding the mean of grouped frequency distribution, $$u_i=$$

A
$$(x_i +a)/h$$

B
$$h(x_i -a)$$

C
$$(x_i -a)/h$$

D
$$(a-x_i )/h$$

Solution

According to the question,
$$x= a+h(f_i u_i/f_i)$$,
Above formula is a step deviation formula.
In the above formula,
$$x_i$$ is data values,
$$a$$ is assumed mean,
$$h$$ is class size,
When class size is same we simplify the calculations of the mean by computing the coded mean of $$u_1, u_2, u_3.....$$,
Where $$u_i= (x_i -a)/h$$
Hence, option (C) is correct.
Question 6
1 / -0

The marks obtained by $$19$$ students of a class are $$27, 36, 22, 31, 25, 26, 33, 24, 37, 32, 29, 28, 36, 35, 27, 26, 32, 35$$ and $$28$$. Find the upper quartile.

A
$$35$$

B
$$36$$

C
$$37$$

D
None of these

Solution

Arrange the marks in increasing order. we get,
$$22,24,25,26,26,27,27,28,28,29,31,32,32,33,35,35,36,36,37$$

Since, total no. of terms $$(n)=19$$

Upper quartile (Q3)$$=3( \cfrac {n+1}{4})^{th}$$term
$$=3(\cfrac {19+1}{4})^{th}$$term
$$=3\times 5=15^{th}$$term$$=35$$
Question 7
1 / -0

Consider the following data:
$$X$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
$$f$$ $$3$$ $$5$$ $$9$$ - $$2$$
The arithmetic mean of the above distribution is $$2.96$$. What is the missing frequency?

A
$$4$$

B
$$6$$

C
$$7$$

D
$$8$$
Question 8
1 / -0

Find the value of $$n$$, if the median of the numbers $$n + 3, n + 7, .... n + 31, n + 35$$ is $$30$$.

A
$$11$$

B
$$14$$

C
$$19$$

D
$$27$$

E
$$32$$

Solution

Given series is $$n+35 = n+3 + (t-1)4$$
$$\therefore 35=3+4t-4$$
$$\therefore 36=4t$$
$$\therefore t = 9$$ (where $$t$$ is number of terms)
Therefore the middle term is $$5^{th}$$ term
$$5^{th}$$ term is $$ n+3 + 16 = 30$$
$$\therefore n+19=30$$
$$\therefore n=11$$
Question 9
1 / -0

The variate of a distribution takes the values $$1, 2, 3, ...n$$ with frequencies $$n, n -1, n -2,... 3, 2, 1.$$ then mean value of the distribution is

A
$$ \displaystyle \frac{n\left ( n+2 \right )}{3} $$

B
$$ \displaystyle \frac{n\left ( n+1 \right )\left ( n+2 \right )}{6} $$

C
$$ \displaystyle\frac{n+2}{3} $$

D
$$ \displaystyle\frac{\left ( n+1 \right )\left ( n+2 \right )}{6} $$

Solution

$$\displaystyle x_{i}=1, 2,3, 4 ...,n-1, n $$
$$\displaystyle f_{i}=n,n-1,n-2,n-3,...2,1$$
Now $$\displaystyle \sum_{i=1}^n x_if_i=

\sum_{i=1}^{n} [ n+2\left ( n-1 \right )+3\left ( n-2 \right

)...$$ $$\displaystyle +\left ( n-2 \right )2+n.1 ] $$ $$\displaystyle

=\sum_{r=1}^{n}[ ( n+1 )r-r^{2}]$$$$\displaystyle = \left ( n+1 \right

)\sum_{r=1}^{n}r-\sum_{r=1}^{n}r^{2}$$ $$\displaystyle =\frac{\left (

n+1 \right )\left ( n \right )\left ( n+1 \right )}{2}-\frac{n\left (

n+1 \right )\left ( 2n+1 \right )}{6}$$$$\displaystyle = \frac{n\left (

n+1 \right )\left ( n+2 \right )}{6}$$

Also $$\displaystyle \sum f_{i}=

1+2+...+n=\frac{n\left ( n+1 \right )}{2}$$

$$\displaystyle \therefore

\mbox{Mean}=\frac{\sum f_{i}x_{i}}{\sum f_{i}}=\frac{n\left ( n+1 \right )\left

( n+2 \right )}{6}\times \frac{2}{n\left ( n+1 \right

)}$$ $$\displaystyle = \frac{n+2}{3}$$
Question 10
1 / -0

$$X$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$
Frequency $$4$$ $$f$$ $$9$$ $$g$$ $$4$$
The table above gives the frequency distribution of a discrete variable $$X$$ with two missing frequencies. If the total frequency is $$25$$ and the arithmetic mean of $$X$$ is $$2$$, then what is the value of the missing frequency $$f$$?

A
$$4$$

B
$$5$$

C
$$6$$

D
$$7$$

Solution

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$$X$$	$$0$$	$$1$$	$$2$$	$$3$$	$$4$$
Frequency	$$4$$	$$f$$	$$9$$	$$g$$	$$4$$

Age (in years)	14	15	16	17	18
Number of girls	5	8	15	10	2

Measures of Central Tendency Test - 50

Result

Measures of Central Tendency Test - 50

Score

-

Rank

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SHARING IS CARING

Question 1

$$109$$

$$106$$

$$150$$

$$160$$

Solution

Question 2

$$1$$

$$2$$

$$3$$

$$5$$

Solution

Question 3

13.9

14.9

15.9

16.9

Solution

Question 4

Mean and mode

Mean and Median

Mode and Median

Mean,Median and Mode

Solution

Question 5

$$(x_i +a)/h$$

$$h(x_i -a)$$

$$(x_i -a)/h$$

$$(a-x_i )/h$$

Solution

Question 6

$$35$$

$$36$$

$$37$$

None of these

Solution

Question 7

$$4$$

$$6$$

$$7$$

$$8$$

Question 8

$$11$$

$$14$$

$$19$$

$$27$$

$$32$$

Solution

Question 9

$$ \displaystyle \frac{n\left ( n+2 \right )}{3} $$

$$ \displaystyle \frac{n\left ( n+1 \right )\left ( n+2 \right )}{6} $$

$$ \displaystyle\frac{n+2}{3} $$

$$ \displaystyle\frac{\left ( n+1 \right )\left ( n+2 \right )}{6} $$

Solution

Question 10

$$4$$

$$5$$

$$6$$

$$7$$

Solution

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